What are the assumptions about time in the general cross lagged panel model (GCLPM)?
time prgoresses in discrete steps
equally spaced measurement occasions
subjects are measured at the same time points
What are limitations of treating time as a discrete variable when modeling longitudinal data?
biased parameter estimates due to:
unequally spaced measurement waves in the panel data
measurement of individual subjects at each wave spread out over time
uncomparable parameter estimates across studies
Multivariate stochastic differential equation: \[\begin{gather*} \mathrm{d} \boldsymbol{\eta}_i(t)=\left({\mathbf{A}} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t) \end{gather*}\]
Multivariate stochastic equation: \[\begin{gather*} \color{red}{\mathrm{d}} \boldsymbol{\eta}_i(\color{red}{t})=\left(\mathbf{A} \boldsymbol{\eta}_i(\color{red}{t})+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \color{red}{\mathrm{d} t}+\mathbf{G} \color{red}{\mathrm{d}} \mathbf{W}_i(\color{red}{t}) \end{gather*}\] modelling change in latent variables \(\boldsymbol{\eta}_i(t)\) as a function of time over infinitesimal small time steps \(\color{red}{\mathrm{d} t}\)
Multivariate stochastic differential equation: \[\begin{gather*} \mathrm{d} \boldsymbol{\eta}_i(t)=\left(\color{red}{\mathbf{A}} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t) \end{gather*}\] modelling change as a function of time over infinitesimal small time steps \(\mathrm{d} t\)
\(\color{red}{\mathbf{A}}\): drift matrix with auto effects on the diagonal and cross effects on the off-diagonals (\(\mathbf{A} \in \mathbb{R}^{v \times v}\)).
Multivariate stochastic differential equation: \[\begin{gather*} \mathrm{d} \boldsymbol{\eta}_i(t)=\left(\mathbf{A} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t) \end{gather*}\] modelling change as a function of time over infinitesimal small time steps \(\mathrm{d} t\)
\(\mathbf{A}\): drift matrix with auto effects on the diagonal and cross effects on the off-diagonals (\(\mathbf{A} \in \mathbb{R}^{v \times v}\)).
\(\mathbf{z_i}\): time independent predictors
\(\boldsymbol{\chi}_i(t)\): time dependent predictors
\(\mathbf{W}_i(t)\): stochastic error term (Wiener process)
\(\mathbf{G}\): diffusion matrix
Multivariate stochastic differential equation: \[\begin{gather*} \mathrm{d} \boldsymbol{\eta}_i(t)=\left(\mathbf{A} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t) \end{gather*}\] measurement part: \[\begin{gather*} \mathbf{y}(t)=\boldsymbol{\Lambda} \boldsymbol{\eta}(t)+{\boldsymbol{\tau}}+\boldsymbol{\varepsilon}(t) \end{gather*}\] \[\begin{gather*} \boldsymbol{\varepsilon}(t) \sim \mathrm{N}(\mathbf{0}, \mathbf{Θ}) \end{gather*}\]
\(\boldsymbol{\Lambda}\): factor loadings
\(\boldsymbol{\tau}\): intercepts
\(\boldsymbol{\varepsilon}(t)\): measurement error
ctsem & ctsemOMXctsem & ctsemOMX to handle the mathctsem interfaces with StanctsemOMX interfaces with OpenMxctsemOMXctsemOMX uses wide data with time intervals specified in columns:
Quality_T0 Quality_T1 Quality_T2 dT1 dT2 dT3
1 -8.016139 -8.940450 -8.243518 1 1 1
2 -6.367648 -6.762183 -6.731700 1 1 1
3 -7.927406 -8.518258 -8.907190 1 1 1
4 -5.887097 -7.304229 -5.552340 1 1 1
5 -5.415876 -10.365073 -8.733621 1 1 1
6 -6.408571 -7.434263 -4.577385 1 1 1(time points can also be interindividually varying)
Function to convert to wideformat: